Problem 19 - Thomas Calculus

Question

In Exercises 5–30, determine an appropriate viewing window for the given function and use it to display its graph. $$ f(x)=\frac{x^{2}+2}{x^{2}+1} $$

Step-by-Step Solution

Answer

View between \([-5, 5]\) on the x-axis and \([0.5, 2.5]\) on the y-axis.

Step 1: Analyze the Denominator

Examine the function given: \[ f(x) = \frac{x^{2}+2}{x^{2}+1} \]The denominator \(x^2 + 1\) is always positive since \(x^2 \geq 0\). So, the function is defined for all real numbers.

Step 2: Determine End Behavior

For large values of \(x\), the function \(f(x) = \frac{x^{2}+2}{x^{2}+1}\) simplifies to 1. This is because the \(x^2\) terms dominate as \(|x|\) becomes very large. Since both the numerator and the denominator are quadratic polynomials, the function approaches 1 as \(x\) approaches infinity or negative infinity.

Step 3: Find Key Values and Behavior

Calculate \(f(x)\) at specific points:- At \(x = 0\): \[ f(0) = \frac{0^2 + 2}{0^2 + 1} = 2 \]- As \(x\) increases or decreases, observe the values of \(f(x)\) at \(x = 1\) and \(x = -1\): \[ f(1) = \frac{1^2 + 2}{1^2 + 1} = \frac{3}{2} = 1.5 \] \[ f(-1) = \frac{(-1)^2 + 2}{(-1)^2 + 1} = \frac{3}{2} = 1.5 \]These values suggest that \(f(x)\) peaks at \(x = 0\) and levels off as \(x\) moves away from 0.

Step 4: Configure the Viewing Window

Based on the function behavior and key values:- The x-axis limits should capture symmetry around zero, such as from \(-5\) to \(5\).- The y-axis limits must capture the range of values, around \(1\), particularly focusing on the maximum value at \(x=0\). Set y-axis limits from \(0.5\) to \(2.5\).

Key Concepts

Rational Functions

A rational function is a function represented by the ratio of two polynomials. In our particular case, the function is \( f(x) = \frac{x^2 + 2}{x^2 + 1} \). Rational functions are special because they enjoy properties from both algebraic and analytic worlds. You can often simplify them under various circumstances, especially when analyzing their graph. When you graph a rational function, look for:

• Vertical asymptotes, which occur at the values that make the denominator zero, although for this function, there are no vertical asymptotes because \(x^2 + 1\) is never zero.
• Horizontal or oblique asymptotes, evident when simplifying the function at extreme values of \(x\).

A common feature of rational functions is their "end behavior," which we'll explore next.

End Behavior

End behavior refers to how a function behaves as the input \(x\) approaches positive or negative infinity. This is particularly interesting for rational functions as it helps us predict the horizontal asymptotes of the graph. For the function \( f(x) = \frac{x^2 + 2}{x^2 + 1} \), the highest degree term in both the numerator and the denominator is \(x^2\). This symmetry indicates that as \(|x|\) becomes very large, the function effectively simplifies to \(\frac{x^2}{x^2} = 1\). This means as \(x\) approaches both positive and negative infinity, the function \(f(x)\) tends towards the value 1. No matter how large or small \(x\) becomes, \(f(x)\) will get closer and closer to 1, guiding us about its long-range stability and horizontal behavior.

Viewing Window

Choosing an appropriate viewing window is crucial for graphing functions. It ensures meaningful visualization without unnecessary details. In the case of \( f(x) = \frac{x^2 + 2}{x^2 + 1} \), we aim to highlight the key aspects and range of the function. Based on our analysis, the function peaks at \( x = 0 \) with a value of 2. Beyond this, as \(|x|\) moves away from zero, \( f(x) \) levels off to 1. To capture these behaviors, consider an x-axis range from \(-5\) to \(5\) to visualize symmetry around the origin. The y-axis, meanwhile, should start just below the minimum function value seen, at around 0.5, up to the maximum value of 2.5, ensuring the peak and end behavior are clearly visible without graphing becoming too cluttered with unnecessary details.

Function Behavior Analysis

A thorough function behavior analysis helps in understanding the intricate details of a function beyond simple graphing. Let’s look closer at key behaviors of \( f(x) = \frac{x^2 + 2}{x^2 + 1} \).

• Symmetry: This function is symmetric about the y-axis, evident since \(f(-x) = f(x)\). Even functions like this give neat, mirror-image graphs.
• Critical Points: At \( x = 0\), we see the maximum value, \( f(0) = 2 \). In contrast, at \(x = 1\) and \(x = -1\), the function evaluates to 1.5, signifying a drop from the peak value.
• Asymptotic Behavior: Reinforcing our earlier observation, the end behavior signifies a horizontal asymptote at \( y = 1 \), guiding the bounds of our graph visualization.

Understanding these behaviors leads to better predictions about how the graph should look and behave under large transformations or inputs.